Zero-forcing cycle-rank conjecture for edge metric dimension

Let $G$ be a connected graph. Let $Z(G)$ denote its zero forcing number, let $c(G)=\left\vert E(G)\right\vert-\left\vert V(G)\right\vert+1$ be its cyclomatic number, and let $\mathrm{edim}(G)$ denote its edge metric dimension. The zero-forcing cycle-rank conjecture for edge metric dimension.

$$\mathrm{edim}(G)\leq Z(G)+c(G).$$

This is proposed as the edge-metric counterpart of the vertex metric-dimension conjecture. The paper establishes analogous results for cactus graphs and records the bound for $3$-connected graphs, but the assertion for all connected graphs remains open.